How to hide lamp reflection in EEVEE
I thank God for one interesting and unexpected hack He showed me. This might help someone if your scene falls within the following criteria:

If you have a lamp inside some occluding geometry, AND the shadow of the occluder happens to fall across a mirror surface with a reflection plane applied, THEN you can increase the lamp's "Bleed Bias" (I used 0.05) and the SHADOW will actually erase the lamp from the mirror's reflection. Note this only works if the shadow falls across the mirror surface. Proof below:

With bleed bias at 0:


With bleed bias at 0.05:


With bleed bias at 0.05 and reflection plane on floor. Note how the reflection of the lamp shade improved, but now you lose the ability to see the mirror's reflections in the floor's reflection (no reflection of reflection). EDIT: Technically, from this angle, the floor would not reflect the mirror's reflection but rather the ceiling. Cycles confirms this. However, Eevee is still incorrect since the floor's reflection plane cannot reflect ANYTHING in the mirror's reflection planes.


EDIT: Here is the Blend file including one occluded light (inside the lampshade) and one non-occluded light floating in space. You can see that the shadow of the occluded light only erases itself, NOT the other light floating in space. Each light would need its own occluder.